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Shape and stability of axisymmetric levitated viscous drops

Published online by Cambridge University Press:  25 December 2008

JOHN R. LISTER ,
ALICE B. THOMPSON ,
ANTOINE PERRIOT  and
LAURENT DUCHEMIN
JOHN R. LISTER*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Institute of Theoretical Geophysics, Wilberforce Road, Cambridge CB3 0WA, UK
ALICE B. THOMPSON
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
ANTOINE PERRIOT
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Institute of Theoretical Geophysics, Wilberforce Road, Cambridge CB3 0WA, UK
LAURENT DUCHEMIN
Affiliation:
IRPHE, 9, Rue F. Joliot-Curie, 13384 Marseille Cedex 13, France
*
†Email address for correspondence: [email protected]
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Abstract

We consider levitation of an axisymmetric drop of molten glass above a spherical porous mould through which air is injected at a constant velocity. Owing to the viscosity contrast, the float height for a given shape is established on a much shorter time scale than the subsequent deformation of the drop under gravity, surface tension and the underlying lubrication pressure. Equilibrium shapes, in which an internal hydrostatic pressure is coupled to the external lubrication pressure through the total curvature and the Young–Laplace equation, are determined using a numerical continuation scheme. The set of solution branches is surprisingly complicated and shows a rich bifurcation structure in the parameter space (Bo=ρgV2/3/γ, Ca=μav/γ), where Bo is bond number and Ca is capillary number, ρ and V are the drop density and volume, γ the surface tension, μa the air viscosity and v the injection velocity. The linear stability of equilibria is determined using a boundary-integral representation for drop deformation that factors out the rapid vertical adjustment of the float height. The results give good agreement with time-dependent simulations. For sufficiently large Ca there are intervals of Bo for which there are no stable solutions and, as Ca increases, these intervals grow and merge. The region of stability decreases as the mould radius aM increases with an approximate scaling Ca~aM−5, which imposes practical limitations on the use of this geometry for the manufacture of lenses.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 617 , 25 December 2008 , pp. 167 - 185
Copyright
Copyright © Cambridge University Press 2008

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